Howe Pairs in the Theory of Vertex Algebras

Details Howe Pairs in the Theory of Vertex Algebras Howe Pairs in the Theory of Vertex Algebras

2006-05-07

arxiv.org/abs/math/0605174v3

J. Algebra 317, 111-152 (2007)

Mathematics

Representation Theory

A few typos corrected, final version

Scientific paper

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

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